The present invention relates to the development of an improved arrangement for controlling repetitive phenomena cancellation in an arrangement wherein a plurality of residual repetitive phenomena sensors and a plurality of cancelling actuators are provided. The repetitive phenomena being cancelled in certain cases may be unwanted noise, with microphones and loudspeakers as the repetitive phenomena sensors and cancelling actuators, respectively. The repetitive phenomena being cancelled in certain other cases may be unwanted physical vibrations, with vibration sensors and counter vibration actuators as the repetitive phenomena sensors and cancelling actuators, respectively.
A time domain approach to the noise cancellation problem is presented in a paper by S. J. Elliott, I. M. Strothers, and P. A. Nelson, "A Multiple Error LMS Algorithm and Its Application to the Active control of Sound and Vibration," IEEE Transactions on Accoustics, Speech, and Signal Processing, VOL. ASSP-35, No. 10, October 1987, pp. 1423-1434.
The approach taught in the above paper generates cancellation actuator signals by passing a single reference signal derived from the noise signal through Na FIR filters whose taps are adjusted by a modified version of the LMS algorithm. The assumption that the signals are sampled synchronously with the noise period is not required. In fact, the above approach does not assume that the noise signal has to be periodic in the first part of the paper. However, the above approach does assume that the matrix of impulse responses relating the actuator and sensor signals is known. No suggestions on how to estimate the impulse responses are made.
The frequency domain approach to the interpretation of the problem is presented as follows, as shown in FIG. 5 which is a block diagram of the system:
The system consists of a set of Na actuators driven by a controller that produces a signal C which is a Na.times.1 column vector of complex numbers. A set of Ns sensors measures the sum of the actuator signals and undesired noise. The sensor output is the Ns.times.1 residual vector R which at each harmonic has the form EQU R=V+HC (1)
where
V is a Ns.times.1 column vector of noise components and PA1 H is the Ns.times.Na transfer function matrix between the actuators and sensors at the harmonic of interest.
The problem addressed by the present invention is to choose the actuator signals to minimize the sum of the squared magnitudes of the residual components. Suppose that the actuator signals are currently set to the value C which is not necessarily optimum and that the optimum value is Copt=C+dC. The residual with Copt would be EQU Ro=H (C+dC)+V=(HC+V)+H dC=R+H dC (2)
The problem is to find dC to minimize the sum squared residual EQU Ro@Ro
where @ denotes conjugate transpose. An equivalent statement of the problem is: Find dC so that H dC is the least squares approximation to -R. This problem will be represented by the notation EQU -R==H dC (3)
The solution to the least squares problem has been studied extensively. One approach is to set the derivatives of the sum squared error with respect to the real and imaginary parts of the components of dC equal to 0. This leads to the "normal equations" EQU H@ H dC=-H@R (4)
If the columns of H are linearly independent, the closed form solution for the required change in C is EQU dC=-[H@H].sup.-1 H@R (5)
The present invention provides methods and arrangements for accommodating the interaction between the respective actuators and sensors without requiring a specific pairing of the sensors and actuators as in prior art single point cancellation techniques such as exemplified by U.S. Pat. No. 4,473,906 to Warnaka, U.S. Pat. Nos. 4,677,676 and 4,677,677 to Eriksson, and U.S. Pat. Nos. 4,153,815, 4,417,098 and 4,490,841 to Chaplin. The present invention is also a departure from prior art techniques such as described in the above-mentioned Elliot et al. article and U.S. Pat. No. 4,562,589 to Warnaka which handle interactions between multiple sensors and actuators by using time domain filters which do not provide means to cancel selected harmonics of a repetitive phenomena.